# Properties

 Label 4864.2.a.bs.1.8 Level $4864$ Weight $2$ Character 4864.1 Self dual yes Analytic conductor $38.839$ Analytic rank $1$ Dimension $10$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$4864 = 2^{8} \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 4864.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$38.8392355432$$ Analytic rank: $$1$$ Dimension: $$10$$ Coefficient field: $$\mathbb{Q}[x]/(x^{10} - \cdots)$$ Defining polynomial: $$x^{10} - 2 x^{9} - 23 x^{8} + 44 x^{7} + 167 x^{6} - 266 x^{5} - 491 x^{4} + 460 x^{3} + 546 x^{2} + 56 x - 8$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{6}$$ Twist minimal: no (minimal twist has level 2432) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.8 Root $$2.42960$$ of defining polynomial Character $$\chi$$ $$=$$ 4864.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+1.15181 q^{3} +1.07587 q^{5} +0.238565 q^{7} -1.67333 q^{9} +O(q^{10})$$ $$q+1.15181 q^{3} +1.07587 q^{5} +0.238565 q^{7} -1.67333 q^{9} +2.84249 q^{11} -1.31444 q^{13} +1.23921 q^{15} -4.47279 q^{17} -1.00000 q^{19} +0.274782 q^{21} +1.94338 q^{23} -3.84249 q^{25} -5.38280 q^{27} -8.83944 q^{29} -6.64877 q^{31} +3.27402 q^{33} +0.256666 q^{35} +4.06763 q^{37} -1.51399 q^{39} -2.78211 q^{41} +1.21363 q^{43} -1.80029 q^{45} -10.2695 q^{47} -6.94309 q^{49} -5.15181 q^{51} +8.36231 q^{53} +3.05817 q^{55} -1.15181 q^{57} -9.25916 q^{59} +4.66035 q^{61} -0.399197 q^{63} -1.41417 q^{65} -4.31849 q^{67} +2.23841 q^{69} +2.78069 q^{71} -0.134460 q^{73} -4.42583 q^{75} +0.678119 q^{77} +12.7841 q^{79} -1.17998 q^{81} -13.4028 q^{83} -4.81216 q^{85} -10.1814 q^{87} +9.20650 q^{89} -0.313579 q^{91} -7.65814 q^{93} -1.07587 q^{95} -0.247499 q^{97} -4.75643 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$10q - 4q^{3} + 14q^{9} + O(q^{10})$$ $$10q - 4q^{3} + 14q^{9} - 20q^{11} + 4q^{17} - 10q^{19} + 10q^{25} - 28q^{27} - 8q^{33} - 36q^{35} - 12q^{41} + 4q^{43} + 26q^{49} - 36q^{51} + 4q^{57} - 52q^{59} - 24q^{65} - 12q^{67} + 12q^{73} + 12q^{75} + 34q^{81} - 16q^{83} - 20q^{89} - 60q^{91} - 28q^{97} - 60q^{99} + O(q^{100})$$

## Coefficient data

For each $$n$$ we display the coefficients of the $$q$$-expansion $$a_n$$, the Satake parameters $$\alpha_p$$, and the Satake angles $$\theta_p = \textrm{Arg}(\alpha_p)$$.

Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 0 0
$$3$$ 1.15181 0.664999 0.332500 0.943103i $$-0.392108\pi$$
0.332500 + 0.943103i $$0.392108\pi$$
$$4$$ 0 0
$$5$$ 1.07587 0.481146 0.240573 0.970631i $$-0.422665\pi$$
0.240573 + 0.970631i $$0.422665\pi$$
$$6$$ 0 0
$$7$$ 0.238565 0.0901690 0.0450845 0.998983i $$-0.485644\pi$$
0.0450845 + 0.998983i $$0.485644\pi$$
$$8$$ 0 0
$$9$$ −1.67333 −0.557776
$$10$$ 0 0
$$11$$ 2.84249 0.857044 0.428522 0.903531i $$-0.359034\pi$$
0.428522 + 0.903531i $$0.359034\pi$$
$$12$$ 0 0
$$13$$ −1.31444 −0.364560 −0.182280 0.983247i $$-0.558348\pi$$
−0.182280 + 0.983247i $$0.558348\pi$$
$$14$$ 0 0
$$15$$ 1.23921 0.319962
$$16$$ 0 0
$$17$$ −4.47279 −1.08481 −0.542405 0.840117i $$-0.682486\pi$$
−0.542405 + 0.840117i $$0.682486\pi$$
$$18$$ 0 0
$$19$$ −1.00000 −0.229416
$$20$$ 0 0
$$21$$ 0.274782 0.0599623
$$22$$ 0 0
$$23$$ 1.94338 0.405222 0.202611 0.979259i $$-0.435057\pi$$
0.202611 + 0.979259i $$0.435057\pi$$
$$24$$ 0 0
$$25$$ −3.84249 −0.768499
$$26$$ 0 0
$$27$$ −5.38280 −1.03592
$$28$$ 0 0
$$29$$ −8.83944 −1.64144 −0.820721 0.571329i $$-0.806428\pi$$
−0.820721 + 0.571329i $$0.806428\pi$$
$$30$$ 0 0
$$31$$ −6.64877 −1.19415 −0.597077 0.802184i $$-0.703671\pi$$
−0.597077 + 0.802184i $$0.703671\pi$$
$$32$$ 0 0
$$33$$ 3.27402 0.569933
$$34$$ 0 0
$$35$$ 0.256666 0.0433845
$$36$$ 0 0
$$37$$ 4.06763 0.668715 0.334357 0.942446i $$-0.391481\pi$$
0.334357 + 0.942446i $$0.391481\pi$$
$$38$$ 0 0
$$39$$ −1.51399 −0.242432
$$40$$ 0 0
$$41$$ −2.78211 −0.434492 −0.217246 0.976117i $$-0.569707\pi$$
−0.217246 + 0.976117i $$0.569707\pi$$
$$42$$ 0 0
$$43$$ 1.21363 0.185077 0.0925386 0.995709i $$-0.470502\pi$$
0.0925386 + 0.995709i $$0.470502\pi$$
$$44$$ 0 0
$$45$$ −1.80029 −0.268372
$$46$$ 0 0
$$47$$ −10.2695 −1.49796 −0.748978 0.662595i $$-0.769455\pi$$
−0.748978 + 0.662595i $$0.769455\pi$$
$$48$$ 0 0
$$49$$ −6.94309 −0.991870
$$50$$ 0 0
$$51$$ −5.15181 −0.721398
$$52$$ 0 0
$$53$$ 8.36231 1.14865 0.574326 0.818627i $$-0.305264\pi$$
0.574326 + 0.818627i $$0.305264\pi$$
$$54$$ 0 0
$$55$$ 3.05817 0.412363
$$56$$ 0 0
$$57$$ −1.15181 −0.152561
$$58$$ 0 0
$$59$$ −9.25916 −1.20544 −0.602720 0.797953i $$-0.705916\pi$$
−0.602720 + 0.797953i $$0.705916\pi$$
$$60$$ 0 0
$$61$$ 4.66035 0.596697 0.298349 0.954457i $$-0.403564\pi$$
0.298349 + 0.954457i $$0.403564\pi$$
$$62$$ 0 0
$$63$$ −0.399197 −0.0502941
$$64$$ 0 0
$$65$$ −1.41417 −0.175407
$$66$$ 0 0
$$67$$ −4.31849 −0.527587 −0.263794 0.964579i $$-0.584974\pi$$
−0.263794 + 0.964579i $$0.584974\pi$$
$$68$$ 0 0
$$69$$ 2.23841 0.269472
$$70$$ 0 0
$$71$$ 2.78069 0.330007 0.165003 0.986293i $$-0.447236\pi$$
0.165003 + 0.986293i $$0.447236\pi$$
$$72$$ 0 0
$$73$$ −0.134460 −0.0157373 −0.00786866 0.999969i $$-0.502505\pi$$
−0.00786866 + 0.999969i $$0.502505\pi$$
$$74$$ 0 0
$$75$$ −4.42583 −0.511051
$$76$$ 0 0
$$77$$ 0.678119 0.0772788
$$78$$ 0 0
$$79$$ 12.7841 1.43832 0.719162 0.694843i $$-0.244526\pi$$
0.719162 + 0.694843i $$0.244526\pi$$
$$80$$ 0 0
$$81$$ −1.17998 −0.131109
$$82$$ 0 0
$$83$$ −13.4028 −1.47115 −0.735573 0.677445i $$-0.763087\pi$$
−0.735573 + 0.677445i $$0.763087\pi$$
$$84$$ 0 0
$$85$$ −4.81216 −0.521952
$$86$$ 0 0
$$87$$ −10.1814 −1.09156
$$88$$ 0 0
$$89$$ 9.20650 0.975887 0.487944 0.872875i $$-0.337747\pi$$
0.487944 + 0.872875i $$0.337747\pi$$
$$90$$ 0 0
$$91$$ −0.313579 −0.0328720
$$92$$ 0 0
$$93$$ −7.65814 −0.794112
$$94$$ 0 0
$$95$$ −1.07587 −0.110382
$$96$$ 0 0
$$97$$ −0.247499 −0.0251297 −0.0125648 0.999921i $$-0.504000\pi$$
−0.0125648 + 0.999921i $$0.504000\pi$$
$$98$$ 0 0
$$99$$ −4.75643 −0.478039
$$100$$ 0 0
$$101$$ 16.8026 1.67193 0.835963 0.548786i $$-0.184910\pi$$
0.835963 + 0.548786i $$0.184910\pi$$
$$102$$ 0 0
$$103$$ 5.40957 0.533020 0.266510 0.963832i $$-0.414129\pi$$
0.266510 + 0.963832i $$0.414129\pi$$
$$104$$ 0 0
$$105$$ 0.295631 0.0288506
$$106$$ 0 0
$$107$$ −6.39113 −0.617853 −0.308927 0.951086i $$-0.599970\pi$$
−0.308927 + 0.951086i $$0.599970\pi$$
$$108$$ 0 0
$$109$$ −14.8779 −1.42505 −0.712524 0.701648i $$-0.752448\pi$$
−0.712524 + 0.701648i $$0.752448\pi$$
$$110$$ 0 0
$$111$$ 4.68515 0.444695
$$112$$ 0 0
$$113$$ 8.59073 0.808148 0.404074 0.914726i $$-0.367594\pi$$
0.404074 + 0.914726i $$0.367594\pi$$
$$114$$ 0 0
$$115$$ 2.09083 0.194971
$$116$$ 0 0
$$117$$ 2.19949 0.203343
$$118$$ 0 0
$$119$$ −1.06705 −0.0978163
$$120$$ 0 0
$$121$$ −2.92023 −0.265476
$$122$$ 0 0
$$123$$ −3.20446 −0.288937
$$124$$ 0 0
$$125$$ −9.51342 −0.850906
$$126$$ 0 0
$$127$$ −1.15983 −0.102918 −0.0514592 0.998675i $$-0.516387\pi$$
−0.0514592 + 0.998675i $$0.516387\pi$$
$$128$$ 0 0
$$129$$ 1.39788 0.123076
$$130$$ 0 0
$$131$$ 3.85559 0.336864 0.168432 0.985713i $$-0.446130\pi$$
0.168432 + 0.985713i $$0.446130\pi$$
$$132$$ 0 0
$$133$$ −0.238565 −0.0206862
$$134$$ 0 0
$$135$$ −5.79122 −0.498428
$$136$$ 0 0
$$137$$ −1.42976 −0.122152 −0.0610761 0.998133i $$-0.519453\pi$$
−0.0610761 + 0.998133i $$0.519453\pi$$
$$138$$ 0 0
$$139$$ −3.58190 −0.303813 −0.151907 0.988395i $$-0.548541\pi$$
−0.151907 + 0.988395i $$0.548541\pi$$
$$140$$ 0 0
$$141$$ −11.8285 −0.996139
$$142$$ 0 0
$$143$$ −3.73629 −0.312444
$$144$$ 0 0
$$145$$ −9.51013 −0.789773
$$146$$ 0 0
$$147$$ −7.99713 −0.659592
$$148$$ 0 0
$$149$$ 16.2763 1.33341 0.666705 0.745322i $$-0.267704\pi$$
0.666705 + 0.745322i $$0.267704\pi$$
$$150$$ 0 0
$$151$$ −14.3242 −1.16569 −0.582845 0.812583i $$-0.698061\pi$$
−0.582845 + 0.812583i $$0.698061\pi$$
$$152$$ 0 0
$$153$$ 7.48445 0.605082
$$154$$ 0 0
$$155$$ −7.15325 −0.574563
$$156$$ 0 0
$$157$$ 3.73495 0.298081 0.149041 0.988831i $$-0.452381\pi$$
0.149041 + 0.988831i $$0.452381\pi$$
$$158$$ 0 0
$$159$$ 9.63181 0.763852
$$160$$ 0 0
$$161$$ 0.463621 0.0365385
$$162$$ 0 0
$$163$$ 3.98861 0.312412 0.156206 0.987724i $$-0.450074\pi$$
0.156206 + 0.987724i $$0.450074\pi$$
$$164$$ 0 0
$$165$$ 3.52243 0.274221
$$166$$ 0 0
$$167$$ 8.17158 0.632336 0.316168 0.948703i $$-0.397604\pi$$
0.316168 + 0.948703i $$0.397604\pi$$
$$168$$ 0 0
$$169$$ −11.2722 −0.867096
$$170$$ 0 0
$$171$$ 1.67333 0.127963
$$172$$ 0 0
$$173$$ 0.351356 0.0267131 0.0133565 0.999911i $$-0.495748\pi$$
0.0133565 + 0.999911i $$0.495748\pi$$
$$174$$ 0 0
$$175$$ −0.916684 −0.0692948
$$176$$ 0 0
$$177$$ −10.6648 −0.801616
$$178$$ 0 0
$$179$$ −15.2740 −1.14163 −0.570817 0.821077i $$-0.693373\pi$$
−0.570817 + 0.821077i $$0.693373\pi$$
$$180$$ 0 0
$$181$$ 8.94630 0.664974 0.332487 0.943108i $$-0.392112\pi$$
0.332487 + 0.943108i $$0.392112\pi$$
$$182$$ 0 0
$$183$$ 5.36785 0.396803
$$184$$ 0 0
$$185$$ 4.37626 0.321749
$$186$$ 0 0
$$187$$ −12.7139 −0.929730
$$188$$ 0 0
$$189$$ −1.28415 −0.0934079
$$190$$ 0 0
$$191$$ 18.3946 1.33098 0.665492 0.746405i $$-0.268222\pi$$
0.665492 + 0.746405i $$0.268222\pi$$
$$192$$ 0 0
$$193$$ −5.09902 −0.367035 −0.183518 0.983016i $$-0.558748\pi$$
−0.183518 + 0.983016i $$0.558748\pi$$
$$194$$ 0 0
$$195$$ −1.62886 −0.116645
$$196$$ 0 0
$$197$$ −20.2916 −1.44572 −0.722860 0.690995i $$-0.757173\pi$$
−0.722860 + 0.690995i $$0.757173\pi$$
$$198$$ 0 0
$$199$$ 0.310999 0.0220461 0.0110231 0.999939i $$-0.496491\pi$$
0.0110231 + 0.999939i $$0.496491\pi$$
$$200$$ 0 0
$$201$$ −4.97408 −0.350845
$$202$$ 0 0
$$203$$ −2.10878 −0.148007
$$204$$ 0 0
$$205$$ −2.99320 −0.209054
$$206$$ 0 0
$$207$$ −3.25191 −0.226023
$$208$$ 0 0
$$209$$ −2.84249 −0.196619
$$210$$ 0 0
$$211$$ −14.3284 −0.986406 −0.493203 0.869914i $$-0.664174\pi$$
−0.493203 + 0.869914i $$0.664174\pi$$
$$212$$ 0 0
$$213$$ 3.20283 0.219454
$$214$$ 0 0
$$215$$ 1.30572 0.0890491
$$216$$ 0 0
$$217$$ −1.58616 −0.107676
$$218$$ 0 0
$$219$$ −0.154872 −0.0104653
$$220$$ 0 0
$$221$$ 5.87921 0.395479
$$222$$ 0 0
$$223$$ −12.8621 −0.861312 −0.430656 0.902516i $$-0.641718\pi$$
−0.430656 + 0.902516i $$0.641718\pi$$
$$224$$ 0 0
$$225$$ 6.42976 0.428650
$$226$$ 0 0
$$227$$ −18.4491 −1.22451 −0.612256 0.790659i $$-0.709738\pi$$
−0.612256 + 0.790659i $$0.709738\pi$$
$$228$$ 0 0
$$229$$ −4.73279 −0.312751 −0.156376 0.987698i $$-0.549981\pi$$
−0.156376 + 0.987698i $$0.549981\pi$$
$$230$$ 0 0
$$231$$ 0.781066 0.0513903
$$232$$ 0 0
$$233$$ −4.27809 −0.280267 −0.140133 0.990133i $$-0.544753\pi$$
−0.140133 + 0.990133i $$0.544753\pi$$
$$234$$ 0 0
$$235$$ −11.0487 −0.720735
$$236$$ 0 0
$$237$$ 14.7249 0.956483
$$238$$ 0 0
$$239$$ 20.3905 1.31895 0.659474 0.751727i $$-0.270779\pi$$
0.659474 + 0.751727i $$0.270779\pi$$
$$240$$ 0 0
$$241$$ 25.5877 1.64825 0.824123 0.566410i $$-0.191668\pi$$
0.824123 + 0.566410i $$0.191668\pi$$
$$242$$ 0 0
$$243$$ 14.7893 0.948732
$$244$$ 0 0
$$245$$ −7.46989 −0.477234
$$246$$ 0 0
$$247$$ 1.31444 0.0836358
$$248$$ 0 0
$$249$$ −15.4375 −0.978311
$$250$$ 0 0
$$251$$ −28.9847 −1.82950 −0.914749 0.404024i $$-0.867611\pi$$
−0.914749 + 0.404024i $$0.867611\pi$$
$$252$$ 0 0
$$253$$ 5.52404 0.347293
$$254$$ 0 0
$$255$$ −5.54270 −0.347098
$$256$$ 0 0
$$257$$ 5.10203 0.318256 0.159128 0.987258i $$-0.449132\pi$$
0.159128 + 0.987258i $$0.449132\pi$$
$$258$$ 0 0
$$259$$ 0.970394 0.0602974
$$260$$ 0 0
$$261$$ 14.7913 0.915558
$$262$$ 0 0
$$263$$ 11.6953 0.721160 0.360580 0.932728i $$-0.382579\pi$$
0.360580 + 0.932728i $$0.382579\pi$$
$$264$$ 0 0
$$265$$ 8.99680 0.552669
$$266$$ 0 0
$$267$$ 10.6042 0.648964
$$268$$ 0 0
$$269$$ 0.163433 0.00996466 0.00498233 0.999988i $$-0.498414\pi$$
0.00498233 + 0.999988i $$0.498414\pi$$
$$270$$ 0 0
$$271$$ 1.01216 0.0614846 0.0307423 0.999527i $$-0.490213\pi$$
0.0307423 + 0.999527i $$0.490213\pi$$
$$272$$ 0 0
$$273$$ −0.361184 −0.0218599
$$274$$ 0 0
$$275$$ −10.9223 −0.658637
$$276$$ 0 0
$$277$$ −3.14825 −0.189160 −0.0945800 0.995517i $$-0.530151\pi$$
−0.0945800 + 0.995517i $$0.530151\pi$$
$$278$$ 0 0
$$279$$ 11.1256 0.666071
$$280$$ 0 0
$$281$$ 24.6734 1.47189 0.735946 0.677040i $$-0.236738\pi$$
0.735946 + 0.677040i $$0.236738\pi$$
$$282$$ 0 0
$$283$$ 13.8095 0.820889 0.410445 0.911886i $$-0.365374\pi$$
0.410445 + 0.911886i $$0.365374\pi$$
$$284$$ 0 0
$$285$$ −1.23921 −0.0734042
$$286$$ 0 0
$$287$$ −0.663713 −0.0391777
$$288$$ 0 0
$$289$$ 3.00584 0.176814
$$290$$ 0 0
$$291$$ −0.285072 −0.0167112
$$292$$ 0 0
$$293$$ −8.24682 −0.481784 −0.240892 0.970552i $$-0.577440\pi$$
−0.240892 + 0.970552i $$0.577440\pi$$
$$294$$ 0 0
$$295$$ −9.96169 −0.579992
$$296$$ 0 0
$$297$$ −15.3006 −0.887829
$$298$$ 0 0
$$299$$ −2.55445 −0.147728
$$300$$ 0 0
$$301$$ 0.289530 0.0166882
$$302$$ 0 0
$$303$$ 19.3535 1.11183
$$304$$ 0 0
$$305$$ 5.01396 0.287098
$$306$$ 0 0
$$307$$ −17.4540 −0.996153 −0.498076 0.867133i $$-0.665960\pi$$
−0.498076 + 0.867133i $$0.665960\pi$$
$$308$$ 0 0
$$309$$ 6.23080 0.354458
$$310$$ 0 0
$$311$$ 10.5256 0.596851 0.298425 0.954433i $$-0.403539\pi$$
0.298425 + 0.954433i $$0.403539\pi$$
$$312$$ 0 0
$$313$$ 14.3498 0.811098 0.405549 0.914073i $$-0.367080\pi$$
0.405549 + 0.914073i $$0.367080\pi$$
$$314$$ 0 0
$$315$$ −0.429486 −0.0241988
$$316$$ 0 0
$$317$$ −16.3470 −0.918138 −0.459069 0.888401i $$-0.651817\pi$$
−0.459069 + 0.888401i $$0.651817\pi$$
$$318$$ 0 0
$$319$$ −25.1260 −1.40679
$$320$$ 0 0
$$321$$ −7.36138 −0.410872
$$322$$ 0 0
$$323$$ 4.47279 0.248873
$$324$$ 0 0
$$325$$ 5.05073 0.280164
$$326$$ 0 0
$$327$$ −17.1366 −0.947656
$$328$$ 0 0
$$329$$ −2.44993 −0.135069
$$330$$ 0 0
$$331$$ 15.0118 0.825123 0.412562 0.910930i $$-0.364634\pi$$
0.412562 + 0.910930i $$0.364634\pi$$
$$332$$ 0 0
$$333$$ −6.80649 −0.372993
$$334$$ 0 0
$$335$$ −4.64615 −0.253846
$$336$$ 0 0
$$337$$ 0.462187 0.0251769 0.0125885 0.999921i $$-0.495993\pi$$
0.0125885 + 0.999921i $$0.495993\pi$$
$$338$$ 0 0
$$339$$ 9.89491 0.537418
$$340$$ 0 0
$$341$$ −18.8991 −1.02344
$$342$$ 0 0
$$343$$ −3.32633 −0.179605
$$344$$ 0 0
$$345$$ 2.40824 0.129655
$$346$$ 0 0
$$347$$ −17.2669 −0.926935 −0.463468 0.886114i $$-0.653395\pi$$
−0.463468 + 0.886114i $$0.653395\pi$$
$$348$$ 0 0
$$349$$ 26.1466 1.39960 0.699798 0.714341i $$-0.253273\pi$$
0.699798 + 0.714341i $$0.253273\pi$$
$$350$$ 0 0
$$351$$ 7.07536 0.377655
$$352$$ 0 0
$$353$$ 12.3930 0.659614 0.329807 0.944048i $$-0.393016\pi$$
0.329807 + 0.944048i $$0.393016\pi$$
$$354$$ 0 0
$$355$$ 2.99167 0.158781
$$356$$ 0 0
$$357$$ −1.22904 −0.0650478
$$358$$ 0 0
$$359$$ −14.1664 −0.747673 −0.373837 0.927495i $$-0.621958\pi$$
−0.373837 + 0.927495i $$0.621958\pi$$
$$360$$ 0 0
$$361$$ 1.00000 0.0526316
$$362$$ 0 0
$$363$$ −3.36356 −0.176541
$$364$$ 0 0
$$365$$ −0.144662 −0.00757195
$$366$$ 0 0
$$367$$ 3.00328 0.156770 0.0783850 0.996923i $$-0.475024\pi$$
0.0783850 + 0.996923i $$0.475024\pi$$
$$368$$ 0 0
$$369$$ 4.65538 0.242349
$$370$$ 0 0
$$371$$ 1.99495 0.103573
$$372$$ 0 0
$$373$$ −38.0432 −1.96980 −0.984901 0.173119i $$-0.944615\pi$$
−0.984901 + 0.173119i $$0.944615\pi$$
$$374$$ 0 0
$$375$$ −10.9577 −0.565852
$$376$$ 0 0
$$377$$ 11.6189 0.598404
$$378$$ 0 0
$$379$$ −28.6291 −1.47058 −0.735290 0.677753i $$-0.762954\pi$$
−0.735290 + 0.677753i $$0.762954\pi$$
$$380$$ 0 0
$$381$$ −1.33591 −0.0684407
$$382$$ 0 0
$$383$$ −24.1407 −1.23353 −0.616767 0.787146i $$-0.711558\pi$$
−0.616767 + 0.787146i $$0.711558\pi$$
$$384$$ 0 0
$$385$$ 0.729571 0.0371824
$$386$$ 0 0
$$387$$ −2.03081 −0.103232
$$388$$ 0 0
$$389$$ 16.2953 0.826206 0.413103 0.910684i $$-0.364445\pi$$
0.413103 + 0.910684i $$0.364445\pi$$
$$390$$ 0 0
$$391$$ −8.69231 −0.439589
$$392$$ 0 0
$$393$$ 4.44091 0.224014
$$394$$ 0 0
$$395$$ 13.7541 0.692043
$$396$$ 0 0
$$397$$ 36.8139 1.84764 0.923819 0.382829i $$-0.125050\pi$$
0.923819 + 0.382829i $$0.125050\pi$$
$$398$$ 0 0
$$399$$ −0.274782 −0.0137563
$$400$$ 0 0
$$401$$ −23.6899 −1.18302 −0.591508 0.806299i $$-0.701467\pi$$
−0.591508 + 0.806299i $$0.701467\pi$$
$$402$$ 0 0
$$403$$ 8.73941 0.435341
$$404$$ 0 0
$$405$$ −1.26951 −0.0630827
$$406$$ 0 0
$$407$$ 11.5622 0.573118
$$408$$ 0 0
$$409$$ −0.102363 −0.00506154 −0.00253077 0.999997i $$-0.500806\pi$$
−0.00253077 + 0.999997i $$0.500806\pi$$
$$410$$ 0 0
$$411$$ −1.64681 −0.0812311
$$412$$ 0 0
$$413$$ −2.20891 −0.108693
$$414$$ 0 0
$$415$$ −14.4197 −0.707836
$$416$$ 0 0
$$417$$ −4.12568 −0.202035
$$418$$ 0 0
$$419$$ −25.7326 −1.25712 −0.628560 0.777761i $$-0.716355\pi$$
−0.628560 + 0.777761i $$0.716355\pi$$
$$420$$ 0 0
$$421$$ −18.0014 −0.877334 −0.438667 0.898650i $$-0.644549\pi$$
−0.438667 + 0.898650i $$0.644549\pi$$
$$422$$ 0 0
$$423$$ 17.1842 0.835524
$$424$$ 0 0
$$425$$ 17.1867 0.833675
$$426$$ 0 0
$$427$$ 1.11180 0.0538036
$$428$$ 0 0
$$429$$ −4.30350 −0.207775
$$430$$ 0 0
$$431$$ −27.6392 −1.33133 −0.665667 0.746249i $$-0.731853\pi$$
−0.665667 + 0.746249i $$0.731853\pi$$
$$432$$ 0 0
$$433$$ 38.5219 1.85124 0.925621 0.378451i $$-0.123543\pi$$
0.925621 + 0.378451i $$0.123543\pi$$
$$434$$ 0 0
$$435$$ −10.9539 −0.525198
$$436$$ 0 0
$$437$$ −1.94338 −0.0929643
$$438$$ 0 0
$$439$$ 23.1302 1.10394 0.551971 0.833863i $$-0.313876\pi$$
0.551971 + 0.833863i $$0.313876\pi$$
$$440$$ 0 0
$$441$$ 11.6181 0.553241
$$442$$ 0 0
$$443$$ −19.2436 −0.914290 −0.457145 0.889392i $$-0.651128\pi$$
−0.457145 + 0.889392i $$0.651128\pi$$
$$444$$ 0 0
$$445$$ 9.90505 0.469544
$$446$$ 0 0
$$447$$ 18.7473 0.886716
$$448$$ 0 0
$$449$$ 11.0578 0.521851 0.260926 0.965359i $$-0.415972\pi$$
0.260926 + 0.965359i $$0.415972\pi$$
$$450$$ 0 0
$$451$$ −7.90812 −0.372379
$$452$$ 0 0
$$453$$ −16.4988 −0.775183
$$454$$ 0 0
$$455$$ −0.337372 −0.0158162
$$456$$ 0 0
$$457$$ −4.08310 −0.190999 −0.0954996 0.995429i $$-0.530445\pi$$
−0.0954996 + 0.995429i $$0.530445\pi$$
$$458$$ 0 0
$$459$$ 24.0761 1.12378
$$460$$ 0 0
$$461$$ 7.59017 0.353509 0.176755 0.984255i $$-0.443440\pi$$
0.176755 + 0.984255i $$0.443440\pi$$
$$462$$ 0 0
$$463$$ −37.2637 −1.73179 −0.865894 0.500227i $$-0.833250\pi$$
−0.865894 + 0.500227i $$0.833250\pi$$
$$464$$ 0 0
$$465$$ −8.23920 −0.382084
$$466$$ 0 0
$$467$$ 8.92480 0.412991 0.206495 0.978448i $$-0.433794\pi$$
0.206495 + 0.978448i $$0.433794\pi$$
$$468$$ 0 0
$$469$$ −1.03024 −0.0475720
$$470$$ 0 0
$$471$$ 4.30196 0.198224
$$472$$ 0 0
$$473$$ 3.44974 0.158619
$$474$$ 0 0
$$475$$ 3.84249 0.176306
$$476$$ 0 0
$$477$$ −13.9929 −0.640690
$$478$$ 0 0
$$479$$ 19.6241 0.896650 0.448325 0.893871i $$-0.352021\pi$$
0.448325 + 0.893871i $$0.352021\pi$$
$$480$$ 0 0
$$481$$ −5.34666 −0.243787
$$482$$ 0 0
$$483$$ 0.534005 0.0242981
$$484$$ 0 0
$$485$$ −0.266277 −0.0120910
$$486$$ 0 0
$$487$$ 13.5448 0.613775 0.306887 0.951746i $$-0.400712\pi$$
0.306887 + 0.951746i $$0.400712\pi$$
$$488$$ 0 0
$$489$$ 4.59413 0.207754
$$490$$ 0 0
$$491$$ 9.58255 0.432454 0.216227 0.976343i $$-0.430625\pi$$
0.216227 + 0.976343i $$0.430625\pi$$
$$492$$ 0 0
$$493$$ 39.5369 1.78065
$$494$$ 0 0
$$495$$ −5.11732 −0.230006
$$496$$ 0 0
$$497$$ 0.663374 0.0297564
$$498$$ 0 0
$$499$$ −7.24834 −0.324480 −0.162240 0.986751i $$-0.551872\pi$$
−0.162240 + 0.986751i $$0.551872\pi$$
$$500$$ 0 0
$$501$$ 9.41213 0.420503
$$502$$ 0 0
$$503$$ −2.19200 −0.0977367 −0.0488683 0.998805i $$-0.515561\pi$$
−0.0488683 + 0.998805i $$0.515561\pi$$
$$504$$ 0 0
$$505$$ 18.0775 0.804440
$$506$$ 0 0
$$507$$ −12.9835 −0.576618
$$508$$ 0 0
$$509$$ 13.5765 0.601766 0.300883 0.953661i $$-0.402719\pi$$
0.300883 + 0.953661i $$0.402719\pi$$
$$510$$ 0 0
$$511$$ −0.0320774 −0.00141902
$$512$$ 0 0
$$513$$ 5.38280 0.237656
$$514$$ 0 0
$$515$$ 5.82002 0.256461
$$516$$ 0 0
$$517$$ −29.1909 −1.28381
$$518$$ 0 0
$$519$$ 0.404696 0.0177642
$$520$$ 0 0
$$521$$ −6.57594 −0.288097 −0.144049 0.989571i $$-0.546012\pi$$
−0.144049 + 0.989571i $$0.546012\pi$$
$$522$$ 0 0
$$523$$ 22.2161 0.971443 0.485722 0.874114i $$-0.338557\pi$$
0.485722 + 0.874114i $$0.338557\pi$$
$$524$$ 0 0
$$525$$ −1.05585 −0.0460810
$$526$$ 0 0
$$527$$ 29.7386 1.29543
$$528$$ 0 0
$$529$$ −19.2233 −0.835795
$$530$$ 0 0
$$531$$ 15.4936 0.672366
$$532$$ 0 0
$$533$$ 3.65691 0.158398
$$534$$ 0 0
$$535$$ −6.87605 −0.297278
$$536$$ 0 0
$$537$$ −17.5928 −0.759185
$$538$$ 0 0
$$539$$ −19.7357 −0.850076
$$540$$ 0 0
$$541$$ 22.5004 0.967368 0.483684 0.875243i $$-0.339298\pi$$
0.483684 + 0.875243i $$0.339298\pi$$
$$542$$ 0 0
$$543$$ 10.3045 0.442207
$$544$$ 0 0
$$545$$ −16.0068 −0.685656
$$546$$ 0 0
$$547$$ 24.1551 1.03280 0.516399 0.856348i $$-0.327272\pi$$
0.516399 + 0.856348i $$0.327272\pi$$
$$548$$ 0 0
$$549$$ −7.79830 −0.332823
$$550$$ 0 0
$$551$$ 8.83944 0.376573
$$552$$ 0 0
$$553$$ 3.04984 0.129692
$$554$$ 0 0
$$555$$ 5.04063 0.213963
$$556$$ 0 0
$$557$$ −21.4469 −0.908734 −0.454367 0.890815i $$-0.650135\pi$$
−0.454367 + 0.890815i $$0.650135\pi$$
$$558$$ 0 0
$$559$$ −1.59525 −0.0674717
$$560$$ 0 0
$$561$$ −14.6440 −0.618270
$$562$$ 0 0
$$563$$ −15.3594 −0.647322 −0.323661 0.946173i $$-0.604914\pi$$
−0.323661 + 0.946173i $$0.604914\pi$$
$$564$$ 0 0
$$565$$ 9.24256 0.388837
$$566$$ 0 0
$$567$$ −0.281503 −0.0118220
$$568$$ 0 0
$$569$$ 2.05613 0.0861973 0.0430986 0.999071i $$-0.486277\pi$$
0.0430986 + 0.999071i $$0.486277\pi$$
$$570$$ 0 0
$$571$$ −35.4718 −1.48445 −0.742225 0.670151i $$-0.766229\pi$$
−0.742225 + 0.670151i $$0.766229\pi$$
$$572$$ 0 0
$$573$$ 21.1871 0.885103
$$574$$ 0 0
$$575$$ −7.46741 −0.311413
$$576$$ 0 0
$$577$$ 14.6910 0.611595 0.305798 0.952097i $$-0.401077\pi$$
0.305798 + 0.952097i $$0.401077\pi$$
$$578$$ 0 0
$$579$$ −5.87311 −0.244078
$$580$$ 0 0
$$581$$ −3.19743 −0.132652
$$582$$ 0 0
$$583$$ 23.7698 0.984445
$$584$$ 0 0
$$585$$ 2.36638 0.0978376
$$586$$ 0 0
$$587$$ −1.71339 −0.0707193 −0.0353596 0.999375i $$-0.511258\pi$$
−0.0353596 + 0.999375i $$0.511258\pi$$
$$588$$ 0 0
$$589$$ 6.64877 0.273958
$$590$$ 0 0
$$591$$ −23.3722 −0.961402
$$592$$ 0 0
$$593$$ 20.3947 0.837509 0.418754 0.908100i $$-0.362467\pi$$
0.418754 + 0.908100i $$0.362467\pi$$
$$594$$ 0 0
$$595$$ −1.14801 −0.0470639
$$596$$ 0 0
$$597$$ 0.358212 0.0146607
$$598$$ 0 0
$$599$$ −21.9447 −0.896637 −0.448318 0.893874i $$-0.647977\pi$$
−0.448318 + 0.893874i $$0.647977\pi$$
$$600$$ 0 0
$$601$$ −3.41894 −0.139461 −0.0697306 0.997566i $$-0.522214\pi$$
−0.0697306 + 0.997566i $$0.522214\pi$$
$$602$$ 0 0
$$603$$ 7.22625 0.294276
$$604$$ 0 0
$$605$$ −3.14180 −0.127732
$$606$$ 0 0
$$607$$ 23.1235 0.938555 0.469278 0.883051i $$-0.344514\pi$$
0.469278 + 0.883051i $$0.344514\pi$$
$$608$$ 0 0
$$609$$ −2.42892 −0.0984247
$$610$$ 0 0
$$611$$ 13.4986 0.546095
$$612$$ 0 0
$$613$$ −46.3763 −1.87312 −0.936561 0.350504i $$-0.886010\pi$$
−0.936561 + 0.350504i $$0.886010\pi$$
$$614$$ 0 0
$$615$$ −3.44760 −0.139021
$$616$$ 0 0
$$617$$ −25.2123 −1.01501 −0.507504 0.861649i $$-0.669432\pi$$
−0.507504 + 0.861649i $$0.669432\pi$$
$$618$$ 0 0
$$619$$ 37.2031 1.49532 0.747660 0.664082i $$-0.231178\pi$$
0.747660 + 0.664082i $$0.231178\pi$$
$$620$$ 0 0
$$621$$ −10.4608 −0.419778
$$622$$ 0 0
$$623$$ 2.19635 0.0879948
$$624$$ 0 0
$$625$$ 8.97722 0.359089
$$626$$ 0 0
$$627$$ −3.27402 −0.130752
$$628$$ 0 0
$$629$$ −18.1937 −0.725429
$$630$$ 0 0
$$631$$ −40.8412 −1.62586 −0.812931 0.582360i $$-0.802129\pi$$
−0.812931 + 0.582360i $$0.802129\pi$$
$$632$$ 0 0
$$633$$ −16.5036 −0.655959
$$634$$ 0 0
$$635$$ −1.24783 −0.0495188
$$636$$ 0 0
$$637$$ 9.12627 0.361596
$$638$$ 0 0
$$639$$ −4.65300 −0.184070
$$640$$ 0 0
$$641$$ 5.87446 0.232027 0.116014 0.993248i $$-0.462988\pi$$
0.116014 + 0.993248i $$0.462988\pi$$
$$642$$ 0 0
$$643$$ 21.3344 0.841347 0.420673 0.907212i $$-0.361794\pi$$
0.420673 + 0.907212i $$0.361794\pi$$
$$644$$ 0 0
$$645$$ 1.50394 0.0592176
$$646$$ 0 0
$$647$$ −48.1625 −1.89346 −0.946732 0.322024i $$-0.895637\pi$$
−0.946732 + 0.322024i $$0.895637\pi$$
$$648$$ 0 0
$$649$$ −26.3191 −1.03311
$$650$$ 0 0
$$651$$ −1.82696 −0.0716043
$$652$$ 0 0
$$653$$ 11.3441 0.443929 0.221965 0.975055i $$-0.428753\pi$$
0.221965 + 0.975055i $$0.428753\pi$$
$$654$$ 0 0
$$655$$ 4.14813 0.162081
$$656$$ 0 0
$$657$$ 0.224995 0.00877791
$$658$$ 0 0
$$659$$ −24.6320 −0.959526 −0.479763 0.877398i $$-0.659277\pi$$
−0.479763 + 0.877398i $$0.659277\pi$$
$$660$$ 0 0
$$661$$ −47.3409 −1.84135 −0.920673 0.390335i $$-0.872359\pi$$
−0.920673 + 0.390335i $$0.872359\pi$$
$$662$$ 0 0
$$663$$ 6.77175 0.262993
$$664$$ 0 0
$$665$$ −0.256666 −0.00995308
$$666$$ 0 0
$$667$$ −17.1784 −0.665149
$$668$$ 0 0
$$669$$ −14.8148 −0.572772
$$670$$ 0 0
$$671$$ 13.2470 0.511396
$$672$$ 0 0
$$673$$ −35.1767 −1.35596 −0.677981 0.735079i $$-0.737145\pi$$
−0.677981 + 0.735079i $$0.737145\pi$$
$$674$$ 0 0
$$675$$ 20.6834 0.796103
$$676$$ 0 0
$$677$$ 2.71085 0.104186 0.0520932 0.998642i $$-0.483411\pi$$
0.0520932 + 0.998642i $$0.483411\pi$$
$$678$$ 0 0
$$679$$ −0.0590444 −0.00226592
$$680$$ 0 0
$$681$$ −21.2499 −0.814300
$$682$$ 0 0
$$683$$ 48.4820 1.85511 0.927557 0.373682i $$-0.121905\pi$$
0.927557 + 0.373682i $$0.121905\pi$$
$$684$$ 0 0
$$685$$ −1.53824 −0.0587730
$$686$$ 0 0
$$687$$ −5.45128 −0.207979
$$688$$ 0 0
$$689$$ −10.9917 −0.418752
$$690$$ 0 0
$$691$$ −41.3295 −1.57225 −0.786124 0.618068i $$-0.787915\pi$$
−0.786124 + 0.618068i $$0.787915\pi$$
$$692$$ 0 0
$$693$$ −1.13472 −0.0431043
$$694$$ 0 0
$$695$$ −3.85368 −0.146178
$$696$$ 0 0
$$697$$ 12.4438 0.471342
$$698$$ 0 0
$$699$$ −4.92755 −0.186377
$$700$$ 0 0
$$701$$ 15.9692 0.603147 0.301573 0.953443i $$-0.402488\pi$$
0.301573 + 0.953443i $$0.402488\pi$$
$$702$$ 0 0
$$703$$ −4.06763 −0.153414
$$704$$ 0 0
$$705$$ −12.7260 −0.479288
$$706$$ 0 0
$$707$$ 4.00852 0.150756
$$708$$ 0 0
$$709$$ −1.27532 −0.0478957 −0.0239478 0.999713i $$-0.507624\pi$$
−0.0239478 + 0.999713i $$0.507624\pi$$
$$710$$ 0 0
$$711$$ −21.3920 −0.802263
$$712$$ 0 0
$$713$$ −12.9211 −0.483898
$$714$$ 0 0
$$715$$ −4.01978 −0.150331
$$716$$ 0 0
$$717$$ 23.4860 0.877100
$$718$$ 0 0
$$719$$ −38.6245 −1.44045 −0.720225 0.693740i $$-0.755962\pi$$
−0.720225 + 0.693740i $$0.755962\pi$$
$$720$$ 0 0
$$721$$ 1.29053 0.0480619
$$722$$ 0 0
$$723$$ 29.4722 1.09608
$$724$$ 0 0
$$725$$ 33.9655 1.26145
$$726$$ 0 0
$$727$$ −3.96945 −0.147219 −0.0736095 0.997287i $$-0.523452\pi$$
−0.0736095 + 0.997287i $$0.523452\pi$$
$$728$$ 0 0
$$729$$ 20.5744 0.762015
$$730$$ 0 0
$$731$$ −5.42832 −0.200774
$$732$$ 0 0
$$733$$ 23.9821 0.885800 0.442900 0.896571i $$-0.353950\pi$$
0.442900 + 0.896571i $$0.353950\pi$$
$$734$$ 0 0
$$735$$ −8.60391 −0.317360
$$736$$ 0 0
$$737$$ −12.2753 −0.452165
$$738$$ 0 0
$$739$$ 34.2553 1.26010 0.630051 0.776554i $$-0.283034\pi$$
0.630051 + 0.776554i $$0.283034\pi$$
$$740$$ 0 0
$$741$$ 1.51399 0.0556177
$$742$$ 0 0
$$743$$ 31.0058 1.13749 0.568746 0.822513i $$-0.307429\pi$$
0.568746 + 0.822513i $$0.307429\pi$$
$$744$$ 0 0
$$745$$ 17.5113 0.641564
$$746$$ 0 0
$$747$$ 22.4273 0.820571
$$748$$ 0 0
$$749$$ −1.52470 −0.0557112
$$750$$ 0 0
$$751$$ 36.0417 1.31518 0.657590 0.753376i $$-0.271576\pi$$
0.657590 + 0.753376i $$0.271576\pi$$
$$752$$ 0 0
$$753$$ −33.3849 −1.21661
$$754$$ 0 0
$$755$$ −15.4111 −0.560867
$$756$$ 0 0
$$757$$ 26.6253 0.967712 0.483856 0.875148i $$-0.339236\pi$$
0.483856 + 0.875148i $$0.339236\pi$$
$$758$$ 0 0
$$759$$ 6.36265 0.230950
$$760$$ 0 0
$$761$$ −40.0064 −1.45023 −0.725115 0.688628i $$-0.758213\pi$$
−0.725115 + 0.688628i $$0.758213\pi$$
$$762$$ 0 0
$$763$$ −3.54935 −0.128495
$$764$$ 0 0
$$765$$ 8.05233 0.291133
$$766$$ 0 0
$$767$$ 12.1706 0.439455
$$768$$ 0 0
$$769$$ 19.1147 0.689295 0.344648 0.938732i $$-0.387998\pi$$
0.344648 + 0.938732i $$0.387998\pi$$
$$770$$ 0 0
$$771$$ 5.87658 0.211640
$$772$$ 0 0
$$773$$ 42.2244 1.51871 0.759353 0.650679i $$-0.225516\pi$$
0.759353 + 0.650679i $$0.225516\pi$$
$$774$$ 0 0
$$775$$ 25.5479 0.917706
$$776$$ 0 0
$$777$$ 1.11771 0.0400977
$$778$$ 0 0
$$779$$ 2.78211 0.0996793
$$780$$ 0 0
$$781$$ 7.90408 0.282830
$$782$$ 0 0
$$783$$ 47.5809 1.70040
$$784$$ 0 0
$$785$$ 4.01833 0.143421
$$786$$ 0 0
$$787$$ 43.6009 1.55420 0.777102 0.629375i $$-0.216689\pi$$
0.777102 + 0.629375i $$0.216689\pi$$
$$788$$ 0 0
$$789$$ 13.4707 0.479571
$$790$$ 0 0
$$791$$ 2.04945 0.0728699
$$792$$ 0 0
$$793$$ −6.12575 −0.217532
$$794$$ 0 0
$$795$$ 10.3626 0.367524
$$796$$ 0 0
$$797$$ −14.6901 −0.520351 −0.260176 0.965561i $$-0.583780\pi$$
−0.260176 + 0.965561i $$0.583780\pi$$
$$798$$ 0 0
$$799$$ 45.9332 1.62500
$$800$$ 0 0
$$801$$ −15.4055 −0.544327
$$802$$ 0 0
$$803$$ −0.382201 −0.0134876
$$804$$ 0 0
$$805$$ 0.498799 0.0175803
$$806$$ 0 0
$$807$$ 0.188244 0.00662649
$$808$$ 0 0
$$809$$ −4.76004 −0.167354 −0.0836771 0.996493i $$-0.526666\pi$$
−0.0836771 + 0.996493i $$0.526666\pi$$
$$810$$ 0 0
$$811$$ 14.6172 0.513279 0.256640 0.966507i $$-0.417385\pi$$
0.256640 + 0.966507i $$0.417385\pi$$
$$812$$ 0 0
$$813$$ 1.16582 0.0408872
$$814$$ 0 0
$$815$$ 4.29125 0.150316
$$816$$ 0 0
$$817$$ −1.21363 −0.0424596
$$818$$ 0 0
$$819$$ 0.524721 0.0183352
$$820$$ 0 0
$$821$$ −32.3079 −1.12755 −0.563776 0.825928i $$-0.690652\pi$$
−0.563776 + 0.825928i $$0.690652\pi$$
$$822$$ 0 0
$$823$$ −9.79982 −0.341600 −0.170800 0.985306i $$-0.554635\pi$$
−0.170800 + 0.985306i $$0.554635\pi$$
$$824$$ 0 0
$$825$$ −12.5804 −0.437993
$$826$$ 0 0
$$827$$ 11.2066 0.389691 0.194845 0.980834i $$-0.437579\pi$$
0.194845 + 0.980834i $$0.437579\pi$$
$$828$$ 0 0
$$829$$ −45.4323 −1.57793 −0.788965 0.614438i $$-0.789383\pi$$
−0.788965 + 0.614438i $$0.789383\pi$$
$$830$$ 0 0
$$831$$ −3.62619 −0.125791
$$832$$ 0 0
$$833$$ 31.0550 1.07599
$$834$$ 0 0
$$835$$ 8.79160 0.304246
$$836$$ 0 0
$$837$$ 35.7890 1.23705
$$838$$ 0 0
$$839$$ −16.8997 −0.583441 −0.291721 0.956504i $$-0.594228\pi$$
−0.291721 + 0.956504i $$0.594228\pi$$
$$840$$ 0 0
$$841$$ 49.1356 1.69433
$$842$$ 0 0
$$843$$ 28.4191 0.978807
$$844$$ 0 0
$$845$$ −12.1275 −0.417200
$$846$$ 0 0
$$847$$ −0.696664 −0.0239377
$$848$$ 0 0
$$849$$ 15.9059 0.545890
$$850$$ 0 0
$$851$$ 7.90494 0.270978
$$852$$ 0 0
$$853$$ −37.7943 −1.29405 −0.647027 0.762467i $$-0.723988\pi$$
−0.647027 + 0.762467i $$0.723988\pi$$
$$854$$ 0 0
$$855$$ 1.80029 0.0615687
$$856$$ 0 0
$$857$$ −38.8619 −1.32750 −0.663749 0.747956i $$-0.731036\pi$$
−0.663749 + 0.747956i $$0.731036\pi$$
$$858$$ 0 0
$$859$$ 41.1795 1.40503 0.702513 0.711671i $$-0.252061\pi$$
0.702513 + 0.711671i $$0.252061\pi$$
$$860$$ 0 0
$$861$$ −0.764473 −0.0260532
$$862$$ 0 0
$$863$$ −25.5909 −0.871125 −0.435562 0.900159i $$-0.643450\pi$$
−0.435562 + 0.900159i $$0.643450\pi$$
$$864$$ 0 0
$$865$$ 0.378015 0.0128529
$$866$$ 0 0
$$867$$ 3.46216 0.117581
$$868$$ 0 0
$$869$$ 36.3387 1.23271
$$870$$ 0 0
$$871$$ 5.67639 0.192337
$$872$$ 0 0
$$873$$ 0.414146 0.0140167
$$874$$ 0 0
$$875$$ −2.26957 −0.0767253
$$876$$ 0 0
$$877$$ 14.4790 0.488920 0.244460 0.969659i $$-0.421389\pi$$
0.244460 + 0.969659i $$0.421389\pi$$
$$878$$ 0 0
$$879$$ −9.49879 −0.320386
$$880$$ 0 0
$$881$$ 48.3407 1.62864 0.814320 0.580416i $$-0.197110\pi$$
0.814320 + 0.580416i $$0.197110\pi$$
$$882$$ 0 0
$$883$$ 17.2669 0.581077 0.290539 0.956863i $$-0.406166\pi$$
0.290539 + 0.956863i $$0.406166\pi$$
$$884$$ 0 0
$$885$$ −11.4740 −0.385694
$$886$$ 0 0
$$887$$ 51.0650 1.71460 0.857298 0.514820i $$-0.172141\pi$$
0.857298 + 0.514820i $$0.172141\pi$$
$$888$$ 0 0
$$889$$ −0.276695 −0.00928006
$$890$$ 0 0
$$891$$ −3.35410 −0.112366
$$892$$ 0 0
$$893$$ 10.2695 0.343655
$$894$$ 0 0
$$895$$ −16.4329 −0.549292
$$896$$ 0 0
$$897$$ −2.94225 −0.0982388
$$898$$ 0 0
$$899$$ 58.7714 1.96014
$$900$$ 0 0
$$901$$ −37.4028 −1.24607
$$902$$ 0 0
$$903$$ 0.333484 0.0110977
$$904$$ 0 0
$$905$$ 9.62510 0.319949
$$906$$ 0 0
$$907$$ 30.0974 0.999367 0.499684 0.866208i $$-0.333450\pi$$
0.499684 + 0.866208i $$0.333450\pi$$
$$908$$ 0 0
$$909$$ −28.1164 −0.932561
$$910$$ 0 0
$$911$$ −9.15413 −0.303290 −0.151645 0.988435i $$-0.548457\pi$$
−0.151645 + 0.988435i $$0.548457\pi$$
$$912$$ 0 0
$$913$$ −38.0973 −1.26084
$$914$$ 0 0
$$915$$ 5.77514 0.190920
$$916$$ 0 0
$$917$$ 0.919807 0.0303747
$$918$$ 0 0
$$919$$ −7.49270 −0.247161 −0.123581 0.992335i $$-0.539438\pi$$
−0.123581 + 0.992335i $$0.539438\pi$$
$$920$$ 0 0
$$921$$ −20.1037 −0.662440
$$922$$ 0 0
$$923$$ −3.65505 −0.120307
$$924$$ 0 0
$$925$$ −15.6299 −0.513906
$$926$$ 0 0
$$927$$ −9.05198 −0.297306
$$928$$ 0 0
$$929$$ −12.0792 −0.396306 −0.198153 0.980171i $$-0.563494\pi$$
−0.198153 + 0.980171i $$0.563494\pi$$
$$930$$ 0 0
$$931$$ 6.94309 0.227550
$$932$$ 0 0
$$933$$ 12.1235 0.396905
$$934$$ 0 0
$$935$$ −13.6785 −0.447336
$$936$$ 0 0
$$937$$ 16.6732 0.544690 0.272345 0.962200i $$-0.412201\pi$$
0.272345 + 0.962200i $$0.412201\pi$$
$$938$$ 0 0
$$939$$ 16.5283 0.539380
$$940$$ 0 0
$$941$$ −29.5263 −0.962529 −0.481265 0.876575i $$-0.659822\pi$$
−0.481265 + 0.876575i $$0.659822\pi$$
$$942$$ 0 0
$$943$$ −5.40668 −0.176066
$$944$$ 0 0
$$945$$ −1.38158 −0.0449428
$$946$$ 0 0
$$947$$ −24.2099 −0.786717 −0.393358 0.919385i $$-0.628687\pi$$
−0.393358 + 0.919385i $$0.628687\pi$$
$$948$$ 0 0
$$949$$ 0.176739 0.00573720
$$950$$ 0 0
$$951$$ −18.8287 −0.610561
$$952$$ 0 0
$$953$$ 18.5304 0.600258 0.300129 0.953899i $$-0.402970\pi$$
0.300129 + 0.953899i $$0.402970\pi$$
$$954$$ 0 0
$$955$$ 19.7903 0.640398
$$956$$ 0 0
$$957$$ −28.9405 −0.935513
$$958$$ 0 0
$$959$$ −0.341089 −0.0110143
$$960$$ 0 0
$$961$$ 13.2062 0.426005
$$962$$ 0 0
$$963$$ 10.6945 0.344624
$$964$$ 0 0
$$965$$ −5.48590 −0.176597
$$966$$ 0 0
$$967$$ 54.2821 1.74559 0.872797 0.488084i $$-0.162304\pi$$
0.872797 + 0.488084i $$0.162304\pi$$
$$968$$ 0 0
$$969$$ 5.15181 0.165500
$$970$$ 0 0
$$971$$ −7.69284 −0.246875 −0.123437 0.992352i $$-0.539392\pi$$
−0.123437 + 0.992352i $$0.539392\pi$$
$$972$$ 0 0
$$973$$ −0.854516 −0.0273945
$$974$$ 0 0
$$975$$ 5.81749 0.186309
$$976$$ 0 0
$$977$$ −32.8587 −1.05124 −0.525621 0.850719i $$-0.676167\pi$$
−0.525621 + 0.850719i $$0.676167\pi$$
$$978$$ 0 0
$$979$$ 26.1694 0.836378
$$980$$ 0 0
$$981$$ 24.8957 0.794858
$$982$$ 0 0
$$983$$ −10.4144 −0.332169 −0.166084 0.986112i $$-0.553112\pi$$
−0.166084 + 0.986112i $$0.553112\pi$$
$$984$$ 0 0
$$985$$ −21.8313 −0.695602
$$986$$ 0 0
$$987$$ −2.82186 −0.0898209
$$988$$ 0 0
$$989$$ 2.35855 0.0749974
$$990$$ 0 0
$$991$$ 62.1164 1.97319 0.986596 0.163183i $$-0.0521761\pi$$
0.986596 + 0.163183i $$0.0521761\pi$$
$$992$$ 0 0
$$993$$ 17.2908 0.548706
$$994$$ 0 0
$$995$$ 0.334596 0.0106074
$$996$$ 0 0
$$997$$ 5.60362 0.177468 0.0887341 0.996055i $$-0.471718\pi$$
0.0887341 + 0.996055i $$0.471718\pi$$
$$998$$ 0 0
$$999$$ −21.8952 −0.692735
Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000

## Twists

By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 4864.2.a.bs.1.8 10
4.3 odd 2 4864.2.a.bt.1.4 10
8.3 odd 2 inner 4864.2.a.bs.1.7 10
8.5 even 2 4864.2.a.bt.1.3 10
16.3 odd 4 2432.2.c.j.1217.7 20
16.5 even 4 2432.2.c.j.1217.8 yes 20
16.11 odd 4 2432.2.c.j.1217.14 yes 20
16.13 even 4 2432.2.c.j.1217.13 yes 20

By twisted newform
Twist Min Dim Char Parity Ord Type
2432.2.c.j.1217.7 20 16.3 odd 4
2432.2.c.j.1217.8 yes 20 16.5 even 4
2432.2.c.j.1217.13 yes 20 16.13 even 4
2432.2.c.j.1217.14 yes 20 16.11 odd 4
4864.2.a.bs.1.7 10 8.3 odd 2 inner
4864.2.a.bs.1.8 10 1.1 even 1 trivial
4864.2.a.bt.1.3 10 8.5 even 2
4864.2.a.bt.1.4 10 4.3 odd 2